Find the curve defined by the equation
\[r = \frac{1}{1 - \cos \theta}.\](A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola

Enter the letter of the correct option.
Answer: From $r = \frac{1}{1 - \cos \theta},$
\[r - r \cos \theta = 1.\]Then $r = 1 + r \cos \theta = x + 1,$ so
\[r^2 = (x + 1)^2 = x^2 + 2x + 1.\]Hence, $x^2 + y^2 = x^2 + 2x + 1,$ so
\[y^2 = 2x + 1.\]This represents the graph of a parabola, so the answer is $\boxed{\text{(C)}}.$

[asy]
unitsize(0.5 cm);

pair moo (real t) {
  real r = 1/(1 - Cos(t));
  return (r*Cos(t), r*Sin(t));
}

path foo = moo(1);
real t;

for (t = 1; t <= 359; t = t + 0.1) {
  foo = foo--moo(t);
}

draw(foo,red);

draw((-4,0)--(4,0));
draw((0,-4)--(0,4));

limits((-4,-4),(4,4),Crop);

label("$r = \frac{1}{1 - \cos \theta}$", (6.5,1.5), red);
[/asy]